ME62: Sparse Hermitian or complex symmetric finite-element system: out-of-core frontal method

To solve one or more sets of sparse Hermitian or complex symmetric linear unassembled finite-element equations, \(\mathbf{AX} = \mathbf{B}\), by the frontal method, optionally holding the matrix factor out-of-core in direct-access files. Numerical pivoting is not performed so for Hermitian matrices it is primarily designed for the positive-definite case. Use is made of high-level BLAS kernels. The coefficient matrix \(\mathbf{A}\) must of the form

\[\mathbf{A} = \sum _{ k=1} ^ m \mathbf{A} ^{(k)} ,\]

with \(\mathbf{A} ^{(k)}\) nonzero only in those rows and columns that correspond to variables in the \(k\)-th element.

The frontal method is a variant of Gaussian elimination and involves the factorization \[\mathbf{A} = \mathbf{PLD} {(\mathbf{PL})} ^ H \mathrm{(Hermitian \; case)},\] or \[\mathbf{A} = \mathbf{PLD} {(\mathbf{PL})} ^ T \mathrm{(symmetric \; case),}\] where \(\mathbf{P}\) is a permutation matrix, \(\mathbf{D}\) is a diagonal matrix, and \(\mathbf{L}\) is a unit lower triangular matrix. The solution process is completed by performing the forward elimination \[(\mathbf{PL})\mathbf{DY}\normalfont = \mathbf{B},\] followed by the back substitution \[(\mathbf{PL}) ^H \mathbf{X}\normalfont = \mathbf{Y}\mathrm{ (Hermitian\; case)}\] or \[(\mathbf{PL}) ^T \mathbf{X}\normalfont = \mathbf{Y} \mathrm{(symmetric\; case).}\] ME62 stores the values of the entries in the factors and their indices separately. A principal feature of ME62 is that, by holding the factors out-of-core, large problems can be solved using a predetermined and relatively small amount of in-core memory. At an intermediate stage of the solution, \(l\) say, the ‘front’ contains those variables associated with one or more of \(\mathbf{A} ^{(k)}\), \(k = 1, 2,..., l\), which are also present in one or more of \(\mathbf{A} ^ {(k)}\), \(k = l+1,..., m\). For efficiency, the user should order the \(\mathbf{A} ^{(k)}\) so that the number of variables in the front (the ‘front size’) is small. For example, a very rectangular grid should be ordered pagewise parallel to the short side of the rectangle. The elements may be preordered using the HSL routine MC63.